Logic and Mathematical Language

Language of Mathematics

• Mathematics relies on effective communication, enabling students to understand and solve problems.
• Familiarity with mathematical language and symbols is crucial due to the unique presentation of ideas.
• Mathematics and English share structural similarities, but math has stricter rules and more complex forms.

Unique Characteristics of Mathematical Language

1. Precision:
   • Mathematical language is precise, allowing clear and unambiguous statements.
   • Terms, symbols, and structures ensure concepts are clearly articulated without misinterpretation.
2. Conciseness:
   • Mathematics conveys complex ideas briefly and efficiently.
   • Symbols, formulas, and notations represent intricate relationships and patterns with minimal effort.
3. Powerfulness:
   • Mathematical language simplifies and expresses highly complex ideas in an accessible way.
   • Abstraction and generalization enable solving real-world problems across fields like physics, economics, and computer science.
4. Non-Temporal:
   • Mathematical language is non-temporal, lacking tenses to convey time. This makes it universally applicable.
5. Vocabulary and Parts of Speech:
   • Mathematical language includes specialized vocabulary and parts of speech such as terms, variables, constants, operators, equations, and expressions.

Graphical Presentation

• Content: Grapheme, lexicon, diagrams, symbolic expressions, natural numbers, and integers.
• Lexicon: Variables, verbal expressions, terms, relations, and symbols.
• Symbols: Auxiliary symbols, grouping symbols, decimal marks, operation process symbols, objects, and concept symbols.
• Verbal Expression: Common words with different meanings in mathematics (e.g., signs, difference, volumes) and discipline-specific terms (e.g., parallelogram, quadrilateral, postulates).
• Structure: English morphology, syntax, phonology, semantics, and pragmatics.
  • Morphology: Structure and construction of words.
  • Syntax: Grammar in phrases and sentences.
  • Phonology: Speech sounds.
  • Semantics: Word meaning.
  • Pragmatics: Use of language in context.

Parts of Speech in Mathematics

• Numbers: Symbols representing quantity (nouns).
• Operation Symbols: Addition, division, conjunction, disjunction (connectives).
• Relation Symbols: Equals, less than, greater than (verbs).
• Grouping Symbols: Parentheses, braces, brackets (associate groups).
• Variables: Letters representing quantities (pronouns).

Mathematical Sentences and Expressions

• Mathematical Sentence: States a complete thought with at least a noun and a verb; can be true or false.
• Mathematical Expression: Does not present a complete thought; involves only mathematical objects or symbols acting as nouns.

Examples

• "Manila is the capital of The Philippines" - Sentence (True).
• "The province of Cabite" - Not a sentence (no verb).
• "The number five is a composite number" - Sentence (False).
• $x + 1$ - Expression (no relational symbol).
• $x^2 - 1 = 3$ - Sentence (True for $x = 2$).

Logic

• Logic is the foundation of mathematics and the language of mathematics.
• It is the study of methods and principles used to distinguish correct from incorrect reasoning.
• Logic focuses on the relationship among statements rather than the content of individual statements.

Proposition

• A proposition is a declarative sentence that is either true or false, but not both.
• Propositions are denoted using small letters or variables (e.g., $p$, $q$, $r$).

Types of Sentences

• Exclamatory Sentence: Not a proposition (expresses feeling).
• Interrogative Sentence: Not a proposition (asks a question).
• Imperative Sentence: Not a proposition (gives a command).

Examples of Propositions

1. "Today is Friday" - Proposition.
2. "Thank God it's Friday" - Not a proposition (exclamatory).
3. "Triangle has three sides" - Proposition.
4. "How many sides does a triangle have?" - Not a proposition (interrogative).
5. $3x - 4 = 2$ when $x = 2$ - Proposition.
6. $3x - 4 = 2$ - Not a proposition (no specific value for $x$).
7. "All Filipinos love to sing" - Proposition.
8. "Go to Cebu via air pacific" - Not a proposition (imperative).

Truth Values

• Truth values are denoted as:
  • True = 1
  • False = 0

Examples

• "Mindanao is an island in The Philippines" - Proposition (Truth value = 1).
• "Find a number which divides your age" - Not a proposition (imperative).
• "My seatmate will get a perfect score in the logic exam" - Proposition (Truth value depends).
• "Welcome to The Philippines" - Not a proposition (exclamatory).
• "What is the domain of the function?" - Not a proposition (interrogative).
• "I am lying" - Proposition (Truth value paradoxical).

Logical Operators

Negation

• The negation of a proposition $P$ is denoted as "not $P$" or $\neg P$.
• It is true if $P$ is false, and false if $P$ is true.

Examples

• Proposition: Claudine is the sister of Gretchen.
  • Negation: Claudine is not the sister of Gretchen.
• Proposition: Today is Monday.
  • Negation: Today is not Monday.
• Proposition: The sun rises in the East.
  • Negation: The sun does not rise in the East.
• Proposition: Every mother has a husband.
  • Negation: It is not true that every mother has a husband.
• Proposition: Isabel is playing tennis.
  • Negation: Isabel is not playing tennis.
• Proposition: There exists a rooster that lays egg.
  * Negation: There does not exist a rooster that lays egg.
• Proposition: Monique is eating pizza.
  • Negation: Monique is not eating pizza.
• Proposition: (Name) is a boxer.
  • Negation: (Name) is not a boxer.

Compound Proposition

• Simple Proposition: A single declarative statement.
• Compound Proposition: Formed from simpler propositions using logical connectors (e.g., or, and, not, if…then, if and only if).

Examples

• "Either logic is fun and interesting or it's boring" - Compound proposition.
• "If you are a grade 11 student, then you are a Filipino" - Compound proposition.
• "Julius Babau is a newscaster" - Simple proposition.
• "J and E Manalo Corporation is a construction firm" - Simple proposition.
• "Ankurthias is a commercial model or a movie queen" - Compound proposition.

Disjunction

• The disjunction of propositions $P$ and $Q$ is denoted as $P \lor Q$ ($P$ or $Q$).
• It is true if at least one of $P$ or $Q$ is true.

Examples

• "Anna will hold a concert in IHMC or Anna will brush her teeth." Symbolic form: J \lor O (JVO).
• "The dog is a mammal or the dougong is a mammal."
  • Symbolic form: S \lor E.
• "Three is greater than zero, then seven plus eight is equal to three."
  • Disjunction statement p or h.
  • Either three is greater than zero or seven plus eight is equal to three.
• Russell is a mathematics teacher. Russell is an engineer. Joseph is an engineer.
  *G and S: Russell is a mathematics teacher or Joseph is an engineer.
  *AVS: Either Russell or Joseph is an engineer.
  Russell is a mathematics teacher or an engineer.

Truth Table for Disjunction

Conjunction

• The conjunction of propositions $P$ and $Q$ is denoted as $P \land Q$ ($P$ and $Q$).
• It is true if and only if both $P$ and $Q$ are true.
• Logical connectors: and, but, while.

Examples

• "Justin is eating pizza and Justin is watching TV."
  • Conjunction statement: Justin is eating pizza and watching TV.
• "Mon is doing his homework and Mon is playing basketball."
  • Conjunction: Mon is doing his homework while playing basketball.
• "Sallah will go to the market and Ken will cook for dinner."
  • Conjunction statement: Salo will go to the market and Ken will cook the dinner.
• "Aldrin will pass the logic exam and Aldrin will fight Randall."

Truth Table for Conjunction

Conditional Statement

• A conditional proposition "if $P$ then $Q$" is denoted as $P \rightarrow Q$.
• $P$ is the premise (hypothesis or antecedent), and $Q$ is the conclusion (consequent).
• The conditional proposition is false only when the premise is true and the conclusion is false.

Examples

• If $\sqrt{5}$ is irrational, then 5 is odd. (If $P$ then $E$).
• If $\sqrt{5}$ is irrational, pi is an algebraic number. (If $P$ then $R$).

Truth Table for Conditional Statement

Other Types of Conditional Statements

• Conditional: If A then B. (A -> B)
• Converse: If B then A. (B -> A)
• Inverse: If not A, then not B. (¬A -> ¬B)
• Contrapositive: If not B then not A. (¬B -> ¬A)

Additional examples

• Nana will give you your allowance and you will clean your room.

Biconditional Statement

• The biconditional proposition $P$ if and only if $Q$ is denoted as $P \leftrightarrow Q$.
• It is true whenever $P$ and $Q$ have the same truth values.

Examples

• 3 is odd if and only if 4 is prime.

Truth Table for Biconditional Statement

Grouping Symbols

• Parentheses, braces, and brackets are used to clarify the structure of logical expressions.

• Parentheses are used with "both…and" and "either…or".

  • Both P or Q and R: $(P \lor Q) \land R$
  • P or both Q and R: $P \lor (Q \land R)$
  • Either P and Q or R: $(P \land Q) \lor R$
  • P and either Q or R: $P \land (Q \lor R)$

Truth Tables

• Truth tables are used to determine the truth value of compound propositions.
• A truth table involving $n$ propositions has $2^n$ rows.

Examples

Complex Examples

• Determining the truth table for $\neg P \land (Q \land \neg R)$.

Tautologies, Contradictions, and Contingencies

• Tautology: A proposition that is always true.
• Contradiction: A proposition that is always false.
• Contingency: A proposition that is neither a tautology nor a contradiction.

Examples

• $P \lor \neg P$ is a tautology.
• $P \land \neg P$ is a contradiction.
• $P \rightarrow \neg P$ is a contingency.

Example Truth Table